Integrals - formulas#
- The indefinite integral is a function.
- Integration is a process opposite to the derivation (differentiation). The integral of f(x) is s(x), if it's derivative reproduces this function:
Function s(x) is sometimes called antiderivative of f(x) or intrinsic function.
- If f(x) is integral of some function, then each function in below form is also it's integral:
where C is arbitrary constant. It's so-called integration constant.
This property results from the fact that derivative from the constant (C) function is equal to 0 at each point.
ⓘ Example: The integral of polynomial is:
because when we calculate it's derivative, then we'll get back this polynomial:
- In contrast to derivatives there are no ready-made formulas that can calculate the integral of any function in routine way. In general, integration requires more sophisticated methods adapted to the specific problem.
- Not every function has its intrinsic function. In other words, there are functions whose integral does not exist.
- Many practical problems e.g. in the field of natural or technical sciences, lead to the need of calculating one or more integrals at some point.
- The equation containing integral from the unknown function is called integral equation.
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